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Part I — Algebra 1
Linear Functions · Equations · Inequalities
01
Algebra★☆☆ Slope of a line
What is the slope of the line that passes through the points (2, 5) and (6, 13)?
🔑 Keyslope = rise over run = (y₂−y₁) ÷ (x₂−x₁)
📖 Explanation
Use the slope formula: m = (y₂ − y₁) / (x₂ − x₁)
m = (13 − 5) / (6 − 2) = 8 / 4 = 2
⚠️ Common mistake: subtracting in the wrong order (x first, y second). Always keep y on top, x on bottom.
02
Algebra★★☆ Slope-intercept form
Which equation represents a line with slope −3 and y-intercept 7?
🔑 Keyy = mx + b → m = slope, b = y-intercept
📖 Explanation
In y = mx + b, m is the slope and b is the y-intercept.
slope = −3 → m = −3 | y-intercept = 7 → b = 7
Answer: y = −3x + 7
⚠️ Don't mix up m and b — m is always the coefficient of x, b is the constant.
03
Algebra★★☆ Solving equations
Solve for x: 3(x − 4) = 2x + 1
🔑 KeyDistribute → Collect like terms → Isolate x
📖 Explanation
Step 1: Distribute → 3x − 12 = 2x + 1
Step 2: Subtract 2x → x − 12 = 1
Step 3: Add 12 → x = 13
⚠️ Most mistakes happen in Step 1 — 3 × (−4) = −12, not +12!
04
Algebra★★☆ Inequalities — direction flip
Solve: −2x + 5 > 11 Which values of x satisfy this inequality?
🔑 KeyMultiply / Divide by NEGATIVE → FLIP the inequality sign
📖 Explanation
Step 1: Subtract 5 → −2x > 6
Step 2: Divide by −2 (⚠️ FLIP!) → x < −3
The critical rule: dividing or multiplying both sides by a negative number reverses the inequality sign.
05
Algebra★★☆ Parallel lines
Which line is parallel to y = 4x − 2?
🔑 KeyParallel = SAME slope, different y-intercept
📖 Explanation
Parallel lines have the same slope. The original slope is 4.
Only y = 4x + 7 has slope = 4 (different b = +7, so not the same line).
⚠️ Choice A (−¼) is the perpendicular slope. Don't confuse them!
06
Algebra★★☆ Standard form → slope
What is the slope of the line 3x + 6y = 12?
🔑 KeyRewrite as y = mx + b → solve for y first
📖 Explanation
Solve for y: 6y = −3x + 12 → y = −½x + 2
Slope = −½
⚠️ Many students write slope = 3 (forgetting to divide by 6). Always isolate y!
07
Algebra★★★ Compound inequality
Solve: −1 ≤ 2x − 3 < 7
🔑 KeyApply same operation to ALL THREE parts simultaneously
📖 Explanation
Add 3 to all parts: 2 ≤ 2x < 10
Divide all by 2: 1 ≤ x < 5
⚠️ Always do the same step to ALL THREE parts of a compound inequality — never split it into two separate problems mid-solve.
08
Algebra★★☆ Function notation
If f(x) = 2x + 3, what is f(−4)?
🔑 Keyf(−4) means "plug in x = −4" — replace x with (−4)
📖 Explanation
f(−4) = 2(−4) + 3 = −8 + 3 = −5
⚠️ Common error: 2 × −4 = −8, not −6. Be careful with negative multiplication!
09
Algebra★★★ Point-slope form
A line passes through (3, −1) with slope 2. What is the y-intercept?
🔑 Keyy − y₁ = m(x − x₁) → then solve for b
📖 Explanation
Use y = mx + b → plug in (x, y) = (3, −1), m = 2:
−1 = 2(3) + b → −1 = 6 + b → b = −7
⚠️ Students often forget to substitute both the x AND y values — use the given point for both!
10
Algebra★★★ Absolute value inequality
Solve: |x − 3| ≤ 5
🔑 Key|A| ≤ k means −k ≤ A ≤ k (AND — sandwich)
📖 Explanation
|x − 3| ≤ 5 means: −5 ≤ x − 3 ≤ 5
Add 3 everywhere: −2 ≤ x ≤ 8
⚠️ Answer C (x ≤ −2 OR x ≥ 8) applies to |A| ≥ k (greater than). For ≤, use the "sandwich" (AND) form!
Part II — Geometry
Triangles · Polygons · Circles
11
Geometry★☆☆ Triangle angle sum
A triangle has angles of 47° and 83°. What is the third angle?
🔑 KeyTriangle angle sum = 180° always
📖 Explanation
Sum of angles = 180°
Third angle = 180 − 47 − 83 = 50°
⚠️ Answer D (130°) = 47 + 83. Students sometimes add the two angles instead of subtracting from 180!
12
Geometry★★☆ Pythagorean theorem
In a right triangle, the two legs are 6 and 8. What is the hypotenuse?
🔑 Keya² + b² = c² → c is always the LONGEST side (hypotenuse)
📖 Explanation
6² + 8² = c² → 36 + 64 = 100 → c = √100 = 10
This is the famous 3-4-5 triple, scaled by 2: (6, 8, 10).
⚠️ D (100) is c² not c — don't forget to take the square root!
13
Geometry★★☆ Interior angles of polygon
What is the sum of the interior angles of a hexagon (6-sided polygon)?
🔑 KeySum = (n − 2) × 180° where n = number of sides
📖 Explanation
Sum = (n − 2) × 180 = (6 − 2) × 180 = 4 × 180 = 720°
⚠️ A (540°) is the answer for a pentagon (5 sides). Make sure to use n = 6!
14
Geometry★★☆ Circle circumference
A circle has a diameter of 10 cm. What is its circumference? (Use π ≈ 3.14)
🔑 KeyC = πd = 2πr | diameter = 2 × radius
📖 Explanation
C = π × d = 3.14 × 10 = 31.4 cm
⚠️ B (62.8) = 2πd — students sometimes use 2πd instead of πd. Remember: C = πd OR C = 2πr — pick ONE formula and use it correctly!
15
Geometry★★☆ Circle area
A circle has a radius of 6. What is its area? (Leave answer in terms of π)
🔑 KeyA = πr² — square the RADIUS (not diameter!)
📖 Explanation
A = πr² = π × 6² = π × 36 = 36π
⚠️ A (12π) = πd (using diameter formula). D (144π) = π × 12² (using diameter squared). Always use the RADIUS in the area formula!
16
Geometry★★★ Exterior angle theorem
An exterior angle of a triangle measures 115°. One of the non-adjacent interior angles is 60°. What is the other non-adjacent interior angle?
🔑 KeyExterior angle = sum of two NON-adjacent interior angles
📖 Explanation
Exterior Angle Theorem: exterior angle = sum of the two non-adjacent interior angles
115 = 60 + x → x = 55°
⚠️ Don't confuse "non-adjacent" — the exterior angle is NOT added to the adjacent interior angle (those two are supplementary = 180°).
17
Geometry★★★ Similar triangles
Two similar triangles have sides in a ratio of 2:5. If the smaller triangle has a perimeter of 24, what is the perimeter of the larger triangle?
🔑 KeySimilar shapes → all LINEAR dimensions scale by the SAME ratio
📖 Explanation
If side ratio = 2:5, then perimeter ratio = 2:5 also.
2/5 = 24/x → x = (24 × 5) / 2 = 60
⚠️ Note: Areas scale by ratio², volumes by ratio³ — but perimeter and length scale by the ratio itself.
18
Geometry★★☆ Area of triangle
A triangle has a base of 10 cm and a height of 7 cm. What is its area?
🔑 KeyA = ½ × base × height (height must be PERPENDICULAR to base)
📖 Explanation
A = ½ × b × h = ½ × 10 × 7 = 35 cm²
⚠️ A (70 cm²) = 10 × 7 — forgetting the ½! A triangle is always half of a parallelogram with the same base and height.
19
Geometry★★★ Arc length
A circle has radius 9 cm. What is the arc length of a sector with a central angle of 80°? (Use π ≈ 3.14)
🔑 KeyArc length = (θ/360) × 2πr — it's a FRACTION of the full circle
📖 Explanation
Arc = (80/360) × 2 × 3.14 × 9 = (2/9) × 56.52 ≈ 12.56 cm
⚠️ Don't forget to multiply by 2πr (full circumference), then take the fraction. Using πr² (area formula) is a very common mix-up!
20
Geometry★★★ Regular polygon — each interior angle
What is the measure of each interior angle of a regular octagon (8 sides)?
🔑 KeyEach angle = (n−2)×180 ÷ n → divide total by number of angles
📖 Explanation
Sum = (8−2) × 180 = 6 × 180 = 1080°
Each angle = 1080 ÷ 8 = 135°
⚠️ A (120°) is for a hexagon (6 sides), C (144°) is for a decagon (10 sides). Each polygon has its own unique angle — memorize by formula, not by guessing!